Numerical methods for the interpolation and approximation of data by spline functions

It is often important in practice to obtain approximate representations of physical data by relatively simple mathematical functions. The approximating functions are usually required to meet certain criteria relating to accuracy and smoothness. In the past, polynomials have frequently been used for this task, but it has long been recognised that there are many types of data set for which polynomial approximations are unsatisfactory in that a very high degree may be required to achieve the required accuracy. Moreover, even if such a polynomial can be computed, it frequently tends to exhibit spurious oscillations not present in the data itself. In an attempt to overcome these difficulties attention has turned in recent years to the use of piecewise polynomials or spline functions. A spline function, or simply a spline, is composed of a set of polynomial arcs, usually of low degree, joined end to end in such a way as to form a smooth function. Splines tend to have greater flexibility than polynomials in the approximation of physical data and much attention has been devoted in the last decade to the theory of splines. The development of robust numerical methods for computing with splines lies, however, lagged somewhat behind the theory. The main objective of this work is the construction and analysis of such methods. In order to obtain efficient and stable methods a representation of splines that is well-conditioned and that results in fast computational schemes is required. Representations in terms of B-splines prove to be eminently suitable and accordingly we study B-splines in some detain and give various algorithms for calculations in which they are involved. When B-splines arc used as a basis for interpolation or least-squares data fitting the resulting linear algebraic systems to be solved for the spline coefficients have a special structure. Stable numerical methods that exploit this structure to the full are presented. Our algorithms are used to obtain spline approximations to a variety of data sets drawn from practical applications. Their performance on these problems illustrates the power of splines over more conventional approximating functions